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A Dynamic Pricing Model under Duopoly Competition Daniel Granot Sauder School of Business, University of British Columbia, 2053 Main Mall, Vancouver, B.C., Canada V6T 1Z2 [email protected],

Frieda Granot Sauder School of Business, University of British Columbia, 2053 Main Mall, Vancouver, B.C., Canada V6T 1Z2 [email protected],

Benny Mantin Sauder School of Business, University of British Columbia, 2053 Main Mall, Vancouver, B.C., Canada V6T 1Z2 [email protected],

We introduce and analyze a multi-period, finite horizon, no-replenishment, dynamic pricing model under duopoly competition between two retailers who sell an identical good. According to our duopoly competition model, a single consumer, whose valuation is uniformly distributed on [0, 1], visits only one of the two retailers in each period. If the price posted by the retailer visited by the consumer is below his valuation, he purchases the good. Otherwise, in the next period the consumer visits the other retailer. We show that, as compared to the corresponding single store monopoly model, under our duopoly competition: (i) prices decline exponentially rather than linearly, (ii) the initial price converges to 0.543, rather than 1, and (iii) the system profit loss due to competition may be up to 29.6% of the monopolistic profit. It is also shown that the pricing policy does not change when the model is extended to N similar consumers. The model is further extended into a more general valuation distribution, into the capacitated case, both under similar and identical consumers, and into the centralized case. Key words : Dynamic Pricing; Revenue Management; Duopoly Competition; Zigzag Competition; Markdowns History :

1.

Introduction

Markdowns, which are used by retailers for various of reasons (Pashigian and Bowen, 1991), became so prevalent over the years that a new coin was termed in the apparel industry - “markdown money”1 . Indeed, as Top of the Net (2002) reports that American retailers are losing more than $200 billion a year due to markdowns2 and 78% of all apparel that is currently sold at American 1

“Markdown money” is the amount stores subtract from suppliers’ checks because, they say, the clothes had to be marked down to sell them (New York Times, 2005ab). 2

An analysis of retail and economic trends by the US Census Bureau and the National Retail Federation (NRF), as cited by Top of the Net (2002). 1

2

Granot, Granot, and Mantin: A Dynamic Pricing Model under Duopoly Competition Article submitted to Management Science; manuscript no. (Please, provide the mansucript number!)

chains such as JCPenny, Sears, and Kohl’s is marked down.3 The ubiquity of markdowns is also reflected from the New York Times (2002) article on pricing: “Like all retailers, Saks [an upscale fashion retailer] faces the problem of deciding when to start marking down prices, and at what prices, so the company can glean as much profit as it can, without discounting so early that the company sells out and alienates customers, or so late that the company looks like a museum for unwanted merchandise”. The question, then, is not whether to markdown, but when and how. There is a vast literature which addresses markdowns, and, in general, dynamic pricing and revenue management, in a finite horizon, fixed inventory (i.e., no replenishment) setting. The various models that were introduced to analyze this setting differ mainly in the manner with which they model demand: consumers’ arrival and their valuations. In Lazear (1986) consumers are myopic and identical, all having a common valuation for the product which is uniformly distributed; Besanko and Winston (1990) have considered the case of N strategic, non-identical consumers, whose valuations are uniformly distributed; in Gallego and van Ryzin (1994) a price-dependent stochastic demand is modelled as a Poisson process; Bitran and Mondschein (1997) have extended Gallego and van Ryzin’s model to the case where customers have heterogeneous valuations, and Zhao and Zheng (2000) have studied a more general model where both the intensity of the customer arrival process and the reservation price distribution are time dependent. For recent extensive reviews of pricing models, the reader is referred, e.g., to Elmaghraby and Keskinocak (2003), Bitran and Caldentey’s (2003), and Chan et al. (2004). Competition has a major effect on pricing. Indeed, as Bitran and Caldentey (2003) conclude: “Including market competition is another important extension to the model. [...] price competition among retailers is today the main driver in their selection of a particular pricing policy”. Similarly, Elmaghraby and Keskinocak (2003) have acknowledged that in the NR-I4 markets literature, competition is one of the missing bricks, and Chan et al. (2004) have admitted that “The reality is that most companies are not monopolies, and many specifically compete on product prices or service differentiation, so these effects need to be considered when establishing coordinated pricing and production policies”. Nevertheless, in spite of the recognition that competition has a significant effect on prices, until quite recently, the effect of competition in the context of revenue management, where inventory 3 4

STS market research study, as cited by Top of the Net (2002).

No Replenishment (NR) of inventory with Independent (I) demand over time, where consumers are either Myopic (M) or Strategic (S).


3

is fixed, has received relatively little attention.5 Recent papers addressing competition include Perakis and Sood (2006), who employ a robust optimization approach to study a multi-period, fixed inventory, dynamic pricing model under competition; Gallego and Hu (2006) who extend Gallego and van Ryzin (1994) to oligopolistic competition by modifying the demand intensity (nonhomogeneous Poisson process) to depend on the prices set by all players; In Talluri (2003), prices are pre-determined but the competing retailers choose to offer only a subset of pre-designed products in each period; Lin and Sidbari (2004) consider a consumer choice model to study a competition between firms who sell substitutable products and face a consumer arrival which follows a Bernoulli process; Friesz et al. (2005) study a competitive model among service providers on a network which incorporates demand dynamics (differential evolutionary game); In Xu and Hopp (2004) retailers set inventory levels in the first stage and compete on prices in the second stage. In their model consumers are homogeneous and their arrival is either piecewise deterministic or follows a Brownian motion; Levin et al. (2006) consider an oligopolistic competition over a finite population of strategic consumers having valuations which follow a random utility model for differentiated products. Our approach to evaluate the effect of competition is different, as elaborated below, and the results and insights we derive are substantial and new. Our main objective in this paper is to incorporate competition in a finite, multi-period, fixed inventory dynamic pricing setting, and to evaluate the effect of competition on prices and profits. In our base model, two profit-maximizing retailers, each of whom can satisfy the entire market, compete over prices, and face a single consumer whose valuation is uniformly distributed over [0, 1]. The myopic6 consumer, who does not have full price information, has to visit the retail stores in order to observe the prices7 . We neglect search costs, but assume that due to these costs the consumer is limited to a single store visit per period. At the store the consumer observes the posted price and if it is below his valuation he purchases the good. Otherwise, he leaves the store and 5

We note, however, that, starting with Bertrand and Edgeworth, price competition has been studied extensively in the economics literature (see, e.g., Tirole (1988), Shapiro (1989), and chapter 8 in Talluri and van Ryzin (2004) and the references therein), in the marketing literature (see, e.g., Moorthy (1993) and Eliashberg and Chatterjee (1985) and also in the International Journal of Research in Marketing’s special issue on competition and marketing (2001)), and in operations (see, e.g., Talluri and van Ryzin (2004), Bitran and Caldentey (2003), and Chan et al. (2004)). 6

One may suggest that due to the popularity of markdowns, consumers learn to expect them, and may also postpone their purchasing decisions even if the posted price is below their valuations. However, it is commonly assumed in the literature that consumers are myopic and buy the good as soon as the price drops below their valuations, since, as Bitran and Mondschein (1997) note, consumers “have partial information about inventories, which prevent them, to some extent, from acting strategically.” Thus, the ubiquity of markdowns coupled with the assumption that consumers are myopic imply that if the price tag exceeds a consumer’s valuation, he will return to check the price in the market in the next period. 7

This is a common assumption in the literature. Bitran and Mondschein (1997), for example, state that “customers [...] have little information about current prices before going to the store”.

4


in the following period visits the other retailer.8 Thus, the consumer keeps visiting the stores in a zigzag manner to check if prices have dropped below his valuation, at which point he purchases the good. We refer to this type of competition as zigzag competition. Such competition exposes the two retailers to the difficulties of pricing correctly in a competitive market. If the price is set too low, potential profit may be lost, while if it is set too high, the product may not be sold. We first demonstrate in this paper that in our base model, wherein there is a single consumer, the effect of zigzag competition is quite dramatic. Indeed, not only are prices strictly decreasing, as in Lazear9 , but they decrease exponentially rather than linearly. Moreover, the initial price is increasing in the selling horizon and converges very fast to 0.543, as compared to 1 in Lazear’s model, and every period, except the last two, it decreases by a factor of almost 0.543. Thus, within very few periods the price in our competition model drops nearly to zero, which may explain the short life cycle of some products in competitive markets. The profit loss of the system due to the introduction of zigzag competition is shown to converge to 0.147, which represents a loss of about 29.6% of the profit, and the profit loss of a monopolist due to the introduction of such competition is shown to converge to 64.78% of his pre-competition profit. Finally, we find that all the results still hold when there are N similar10 consumers. To investigate the robustness of our results, we briefly analyze the effect of zigzag competition when some of our assumptions are modified or extended. More specifically, we consider the following cases: (i) the valuation of the good by consumers has a power distribution on [0, 1], (ii) the system is capacitated in the sense that the retailers cannot satisfy the entire demand in the market, and (iii) consumers are identical rather than similar, i.e., all consumers have the same valuation for the product which is uniformly distributed. We note that the proper evaluation of cases (i) and (ii) has necessitated the analysis of corresponding monopoly case. The analysis of the effect of zigzag competition with a power distribution has revealed that the nature of the exponential decline of prices depends on the parameter, q, −1 < q ≤ ∞, of the distribution, which for q = 0, coincides with the uniform distribution. Specifically, the decline of prices is most severe when q → −1, and it is most moderate, yet exponential when T → ∞, when 8

In an ensuing paper (Granot, Granot, and Mantin, 2006) we analyze the considerably more complex case, wherein the consumer may return to the same store with some arbitrary probability (see §3.3). This return probability can serve as a proxy for the intensity of competition in the market. If it is one, there is no competition and the model is reduced to two local monopolists, while if it is zero, as is the case in the zigzag competition which is considered in this paper, then there is an intense competition in the market. 9

Since Lazear’s model can be perceived as a monopoly facing a single consumer over T periods, we compare our results with those obtained by Lazear. 10

That is, the consumers’ valuations are independently drawn from the same uniform distribution.


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q → ∞, i.e., when the valuations by the consumers are very likely to be very close to one. The analysis of the capacitated case, when the retailers cannot satisfy the entire demand, demonstrates that zigzag competition still has a major impact on prices and profits. However, the decline in prices and profits become more moderate when the system is more constraint, i.e., the retailers can supply a small fraction of the market, or when the number of periods increases. Prices in our zigzag competition model with identical consumers are shown to decline about as much as they do with similar consumers. Yet, a quantitative difference between these two cases is uncovered. Specifically, while there is a unique pure-strategy SPNE for prices in the similar consumers case, when consumers are identical, there are multiple, and, even in some cases a continuum of pure-strategy SPNE. In particular, it is shown that for the two-period case with identical consumers there are precisely two pure-strategy SPNE in prices, both of which are non-symmetric. The analysis of the capacitated case with identical consumers has revealed that in some instances, when the system is very constrained, the profit of the duopoly system may exceed the profit of the single store monopoly. However, by definition, this profit is lower than the profit of a centralized two-store monopoly, which is the right reference for comparison, and, in return, is subsequently studied in the paper. The two-store monopoly is similar to the duopoly competition setting with one main difference: the chain is controlled by a central planner who is setting possibly different prices in both retail stores simultaneously. We find that the profit of the two-store monopoly could be strictly higher than the one-store monopoly when the consumers are identical. Indeed, it is shown that for a fixed number of consumers, the profit of the centralized system is increasing in the number of consumers who encounter the higher price, as long as the number of consumers who encounter the lower price is not zero. By contrast, it is shown that when the consumers are not identical, a centralized two-store monopoly system does not have any advantage over the single-store monopoly. Our main contribution in this paper, asides for our model of zigzag competition, is our ability to evaluate and quantify the effect of competition on prices and profits. Specifically, we show that zigzag competition exerts significant pressure on prices, resulting with an exponentially declining price trajectory, and that its introduction reduces the monopolist profit quite dramatically. Further, the effect of such competition is shown to be robust in the sense that it essentially holds (i) for a large class of product valuation distribution, (ii) when the retailers cannot satisfy the entire market demand, and (iii) for both identical and similar consumers. The rest of the paper is arranged as follows. In Section 2 we introduce and analyze the basic zigzag competition model, first with a single consumer and, subsequently, with N similar consumers.


6

Section 3 considers several extensions to the model: a more general distribution of valuation, capacitated system, and briefly discusses a more general visit pattern of consumers between stores. In Section 4 we study the behavior of the model when consumers are identical rather than similar under both the capacitated and uncapacitated cases. Section 5 investigates the case of a two-store monopoly and Section 6 concludes with a brief summary and some discussion on future work.

2. 2.1.

The Model The Basic Model

Two competing retailers, each of whom has in stock only one unit of an identical good, encounter a single myopic consumer whose valuation for that good is V . That is, the consumer is willing to pay up to V for the good, but no more, and he purchases the good once he observes a price which is below his valuation. The retailers do not know V with certainty, but they do have prior knowledge of the density of V , denoted f (V ), with distribution function F (V ). The two retailers are risk-neutral and seek to maximize their expected profits. They have T periods to sell their good to the consumer, and in each of these periods they can post a new price. The consumer does not have any knowledge about the posted prices in the market in each period. To obtain that information he needs to visit each retail store and observe the posted price. We neglect search costs, but assume that the consumer is limited to a single retail store visit per period. We introduce in this paper a new type of competition between the retailers to be referred to as zigzag competition. In zigzag competition, the retailers compete to sell their product to the single consumer who visits only one store per period. In period t he observes a price Rt at the retailer he visits. If this price is below his valuation, he purchases the good. Otherwise he does not purchase the good, and in the following period he visits the other retailer with certainty.11 Figure 1 demonstrates the zigzagging pattern of the consumer across the two retail stores. We refer to the retailer who encounters (resp., does not encounter) the consumer in the first period as Retailer 1 (resp., 2). Thus, in an odd (resp., even) period, if no purchase has occurred before that period, the consumer visits Retailer 1 (resp., 2). In each period the price posted at the retail store not visited by the consumer is not relevant, and no assumption is needed for a retailer’s knowledge about the other retailer’s posted prices. However, since both retailers solve the same model, they can compute the prices that the other 11

Switching behavior of consumers has been also considered in the operations literature, e.g., by Zhao and Atkins (2003), Netessine and Shumsky (2005), and in Netessine et al. (2006). However, in their models switching from one retailer to another may occur due to stockouts, i.e., a consumer might switch to a competing retailer only if the demanded product at the visited location is out of stock.


Retailer 1

Retailer 2

Price

Price

Period 1:

R1

Not Observed

Period 2:

Not Observed

R2

Period 3:

R3

Not Observed

7

§ Figure 1

The zigzagging pattern and the observed prices

retailer is posting. If the good is not sold in period t at price Rt , the retailers can immediately infer that V < Rt . We denote Retailer 1’s (resp., Retailer 2’s) expected profit-to-go function at period t as πt1 (resp., πt2 ). If t is odd, Retailer 1 is visited by the consumer in that period, and we have: πt1 =Rt [1 − Ft (Rt )] + Rt+2 [1 − Ft+2 (Rt+2 )]Ft+1 (Rt+1 )Ft (Rt ) + · · · b T 2−t c t+2i−1 X Y = Fj (Rj ) , Rt+2i [1 − Ft+2i (Rt+2i )] i=0

(1)

j=t

where the first term in πt1 is the expected profit from a sale in period t. It is equal to the price charged in period t times the probability that the good sells in period t, given the information from period t − 1; the second term is the conditional expected profit from a sale in period t + 2, and so on. Similarly, the expected profit-to-go of Retailer 2, who does not encounter the consumer in that period (since t is odd), can be written as: πt2 =Rt+1 [1 − Ft+1 (Rt+1 )]Ft (Rt ) + Rt+3 [1 − Ft+3 (Rt+3 )]Ft+2 (Rt+2 )Ft+1 (Rt+1 )Ft (Rt ) + · · · b T −t−1 c t+2i 2 X Y = Rt+1+2i [1 − Ft+1+2i (Rt+1+2i )] Fj (Rj ) . i=0

(2)

j=t

The profit-to-go functions from (1) and (2) can be expressed in a recursive form. If t is odd, 1 πt1 = Rt [1 − Ft (Rt )] + πt+1 Ft (Rt )

and

2 πt2 = πt+1 Ft (Rt ).

Thus, Retailer 1 sets prices, at odd periods, so as to: M ax

πt1 = M ax Rt

1 Rt [1 − Ft (Rt )] + πt+1 Ft (Rt ) ,

(3)


8

and similarly, at even periods, Retailer 2 solves: M ax 2.2.

πt2 = M ax Rt

2 Rt [1 − Ft (Rt )] + πt+1 Ft (Rt ) .

Uniformly Distributed Valuation

For simplicity, suppose first that the prior on the valuation V is uniformly distributed between zero and one. By Bayes’ Theorem, this implies that the posterior distribution carried from period t to t + 1 is uniform between zero and Rt , so that Ft+1 (V ) =

V . Rt

To illustrate, we solve the problem, backwards, for the last four periods of the zigzag competition. In Table 1 we display the retail prices for both the zigzag competition model and the monopoly model for T = 4. The price ratio at period t is

Rt . Rt−1

Observe from Table 1 that, under zigzag

competition, prices decline much more rapidly and the initial price is much lower compared to the monopoly model. Zigzag Competition

Monopoly

Price ratio

Retailer 1’s Price

Retailer 2’s Price

πt1

πt2

Price Ratio

Price

Period 1

225 418

R1 ≈ 0.538

Not observed

≈ 0.269

≈ 0.077

4 5

0.8

0.4

Period 2

8 15

Not observed

R2 ≈ 0.287

≈ 0.011R1

≈ 0.266R1

3 4

0.6

0.375R1

Period 3

1 2

R3 ≈ 0.143

Not observed

0.25R2

0.0625R2

2 3

0.4

0.33R2

Period 4

1 2

0.25

1 2

0.2

0.25R3

Table 1

Not observed

R4 ≈ 0.0717

0

πt

Price ratios, prices, and profits under zigzag competition and monopoly settings

Lemma 1. The profit-to-go in each period is an affine function of the price that is set in the previous period. That is, πti = Sti Rt−1 , i = 1, 2, where Sti are scalars. Moreover, at odd (resp., even) periods, the values of St1 and St2 can be expressed recursively as follows: St1 = St1 =

1 St+1 2 )2 4(1−St+1

) and St2 =

2 St+1 1 )2 4(1−St+1

(resp., St2 =

1 1 ) 4(1−St+1

(resp.,

1 2 ) ). 4(1−St+1

Proof. The proof is by induction. From Table 1 it can be verified that the recursions hold for the final periods. Assume the Lemma holds for t + 1, and we will show that it holds for an odd t. Rt 1 1 t t t Rt RRt−1 Now, Retailer 1’s profit-to-go is πt1 = Rt (1 − RRt−1 ) + πt+1 = Rt (1 − RRt−1 ) + St+1 , where Rt−1

the last equality follows from the induction hypothesis. Setting the derivative of πt1 with respect to Rt to zero results with Rt =

1 Rt−1 . 1 2(1 − St+1 )

Using (4) we rewrite πt1 and πt2 to obtain: πt1 = St1 =

1 1 4(1 − St+1 )

(4)

1 1 ) Rt−1 4(1−St+1

and

St2 =

and πt2 =

2 St+1

1 )2 4(1−St+1

2 St+1 , 1 4(1 − St+1 )2

Rt−1 . Thus, (5)


9

and due to the induction hypothesis St1 and St2 are scalars as well. The proof for an even t follows in a similar way. Note that the scalars St1 and St2 have a dimension of sales, and will thus be referred to as the normalized sales-to-go in period t of Retailers 1 and 2, respectively, with respect to the price that is set in the previous period, Rt−1 . Let {s1t } (resp., {s2t }) denote the sequence of normalized sales-to-go of the retailer who encounters (resp., does not encounter) the consumer in period t. That is, for an odd t, s1t = St1 = (resp., s2t = St2 =

s1 t+1

2 4(1−s2 t+1 )

1 4(1−s2 t+1 )

), and for an even t, s1t = St2 (resp., s2t = St1 ). Thus, for all t, s1t =

1 4(1 − s2t+1 )

and s2t =

s1t+1 . 4(1 − s2t+1 )2

(6)

Note that in both {s1t } and {s2t } sequences, as well as all other sequences considered in the sequel, the sequences progress from t = T to t = 1. Thus, e.g., the first element in {s1t } is s1T and the last element is s11 . Lemma 2. The normalized sales-to-go sequences {s1t } and {s2t } are bounded sequences. Specifically, {s1t } (resp., {s2t }) is bounded from below by s1 = 1 √ (19+3 33) 3 1 2 √ √ (19+3 33) 3 +(19+3 33) 3 +4

3 2

1 4

(resp., s2 = 0) and from above by12 s1 = √ 1 1 ≈ 0.271 (resp., s2 = 56 − 61 (19 + 3 33) 3 − 23 1 ≈ 0.0803). √ (19+3 33) 3

Proof. The proof is by induction. The induction base is satisfied for the last two periods as s1T = s1T −1 = 14 , s2T = 0, and s2T −1 = As s1t =

1 , 4(1−s2 t+1 )

Thus, s1t =

1 . 16

Suppose that

1 4

≤ s1t+1 < s1 , 0 ≤ s2t+1 ≤ s2 , for some t.

the lowest value for s1t is obtained if the lower bound value of s2t is used.

1 4(1−s2 t+1 )

≥

1 4(1−0)

=

1 4

= s1 , where the last inequality is satisfied by the induction

hypothesis. The largest value for s1t is obtained if the upper bound value of the sequence of s2t is used. We have s1t =

1 2 ) 4(1−st+1

1

1 √ 2 3 4 1− − 1 6 (19+3 33) − 3

1 5 1 +6 √ (19+3 33) 3

≤

1 , 4(1−s2 )

=

3 2

as follows from the induction hypothesis. Thus, s1t ≤

1 √ (19+3 33) 3 1 2 √ √ (19+3 33) 3 +(19+3 33) 3 +4

= s1t . As required, we have shown

that s1 ≤ s1t ≤ s1 . The proof for the boundedness of {s2t } follows in a similar way.

Proposition 1. The sequences of normalized sales-to-go, {s1t } and {s2t }, are monotone convergent sequences as the number of periods, T , goes to infinity, while t goes to 1. Proof. The proof is based on the Monotone Convergence Theorem. The two sequences of the normalized sales, {s1t } and {s2t }, are bounded from above as follows from Lemma 2. Next, we show that the sequences are increasing. 12

These bounds are obtained by omitting the subscripts from (6) and solving for s1 and s2 .


10

{s1t } is increasing if

s1 t s1 t+1

=

1 2 4s1 t+1 (1−st+1 )

≥ 1, which certainly holds (by Lemma 2) as 0 < 4s1t+1 (1 −

s2t+1 ) ≤ 1. The proof that {s2t } is increasing is by induction. We have that s2T −1 = 0. Assume

s2 t+1

s2 t+2

2 1 s1 t st+1 (1−st+2 ) 1 (1−s2 ) s1 s t+1 t+2 t+1

≥ 1, and we will verify that

s2 t s2 t+1

≥ 1. Notice that s2t =

. Since {s1t } is increasing and by the induction hypothesis

1 s1 t st+1

1−s2 t+1

1−s2 t+2 1−s2 t+1

1 16

. Thus,

≥ 1,

s2 t s2 t+1

≥ s2T = s2 t s2 t+1

≥ 1.

=

Proposition 2. The normalized sales-to-go of the retailer who encounters the consumer in period t is strictly larger than the normalized sales-to-go of the retailer who does not encounter the consumer in that period.13 Proposition 3. The normalized sales-to-go of a monopolist is at least as large as the combined normalized sales-to-go of both retailers in the zigzag competition model. Note that for t = 1 the normalized sales-to-go of the retailers coincide with their expected profits. Therefore, Proposition 3 implies: Corollary 1. A monopolist’s profit is at least as large as the combined profits of both retailers in the zigzag competition model. Let us next consider the sequence of price ratios, {Bt }, defined as Bt ≡

Rt 1 = . Rt−1 2(1 − s2t+1 )

(7)

From (6) and (7) one can observe that s1t = 21 Bt and s2t = 4s1t+1 (s1t )2 . Thus, we find that Bt =

1 2 − (Bt+1 )2 Bt+2

.

(8)

Proposition 4. The sequence of price ratios, {Bt }, is increasing and each element therein is bounded from below by 0.5 and from above by14 0.55. From Proposition 4 we obtain: Theorem 1. Under zigzag competition the prices decrease exponentially. Proof. For every t, Bt =

Rt Rt−1

≤ B1 ≤ 0.55. Thus, since R0 = 1, Rt ≤ (0.55)t−1 .

To illustrate the behavior of the model, consider an example with 20 periods. The price ratios and the prices that are set under zigzag competition in each of the 20 periods and the prices in the monopoly setting are plotted in Figure 2(a), whereas the profit-to-go of the retailers are plotted in Figure 2(b). We observe that: 13 14

Proofs which are not presented in the body of the paper are provided in the appendix.

In Proposition 5 we show that {Bt } converges to 0.543 and thus each element in the sequence is bounded from above by the convergence value.


11

• The price ratio under the zigzag competition does not change much over most of the selling

season. As it is solved recursively, it slightly increases from a value of 0.5 at T = 20 to a value of about 0.543 at t = 1, which is dramatically lower than the initial price set in the corresponding monopoly model,

20 . 21

In other words, the fear of losing a sale due to competition significantly

suppresses the initial price. Since the price ratio is equal to the initial price had that period been the first period of the selling horizon, we conclude that the initial price is decreasing with perishability (measured by the length of the selling horizon), though only marginally. • As proven by Theorem 1, the prices that are set over the selling horizon under zigzag compe-

tition are decreasing exponentially. They approach zero fairly quickly. By contrast, in a monopoly market, prices decline linearly. • The profit-to-go of Retailers 1 and 2 also decrease exponentially with some spikes. As the price

declines quickly, the conditional expected valuation of the consumer declines as well, and there is less profit to be made. Thus, the largest part of the profit is achieved in the early periods, and later periods of the selling season contribute very marginally to the total profit. The spikes occur since a retailer who does not encounter the consumer in period t will encounter him only with some probability in period t + 1. 0.3

1

Monopoly Prices

0.25

Retailer 1

0.6

Price Ratio

0.4

Duopoly Prices

0.2

Profit-to-go

Price, Price Ratio

0.8

0.2 0.15

Retailer 2 0.1 0.05

0

0 1

3

5

7

9

11

13

15

17

19

1

3

t, Period Number

(a)Prices and price ratio Figure 2

5

7 9 11 13 15 t, Period Number

17

19

(b)Retailers’ profit-to-go

A 20-period model: price, price ratio and profit-to-go

To illustrate the last point, let us examine the retailers expected profit over a range of selling horizons, as demonstrated in Figure 3(a), wherein the selling horizon, T , ranges from T = 1 up to T = 50. Observe that Retailer 1’s expected profit equals 0.25 in a single and two-period horizons and then converges quickly to about 0.271. Similarly, Retailer 2’s expected profit equals zero in a single period model (as he does not encounter the consumer at all), rises to 0.0625 in a twoperiod horizon and then converges quickly to about 0.08. Thus, we may conclude that longer selling


12

horizons do not contribute much to the expected profit of each of the retailers. Consequently, the system’s profit converges quickly to about 0.35. It can be observed from Figure 3(b) that the monopolist’s profit ( 2(TT+1) ) converges slower but to a higher value. Thus, the larger the value of T is, for T > 1, the larger is the profit advantage of a monopoly over the the zigzag competitors. Consumer’s surplus measures the difference between the consumer’s valuation and the price charged, if the consumer purchased the good. Thus, under both settings, the consumer’s surplus Pt=T equals t=1 12 (Rt−1 − Rt )2 . Figure 3(c) contrasts the consumer’s surplus under zigzag competition and under monopoly. With longer horizons, the monopolist can more efficiently price the good and extract more of the surplus from the consumer and, therefore, the consumer’s surplus in the monopoly setting is decreasing in T . However, in the zigzag competition the two competing retailers are mainly engaged in price cutting competition and, therefore, the consumer’s surplus first increases from 0.125 to 0.156, and then it slightly decreases (since the initial price also marginally increases) and quickly converges to 0.147.

0.3

0.5

Profit

Profit

0.2 0.15

Duopoly

0.3 0.2

0.1

Profit Loss Retailer 2

0.05

0.1

0

0 0

10

20 30 T, Selling horizon

40

50

(a)Retailers’ profits Figure 3

2.3.

Duopoly

0.4

Consumer Surplus

0.25

0.16

Monopoly

Retailer 1

0.12

0.08

Monopoly

0.04

0 0

10

20 30 T, Selling Horizon

40

50

(b)System’s profit and profit loss

0

10

20 30 40 T, Selling horizon

50

(c)Consumer’s surplus

Profit, profit loss, and consumer surplus in monopoly and duopoly under zigzag competition

Values at Convergence

We first provide sharp bounds for the initial price. Proposition 5. The initial price, R1 , is bounded from below by 21 (if T=1 or 2) and monoton√ 1 1 1 ically converges to 13 (17 + 3 33) 3 − 23 1 − 3 ≈ 0.543 as T → ∞. √ (17+3 33) 3

Recall from Theorem 1 that under zigzag competition prices decrease exponentially. From Proposition 5 the price ratio converges to 0.543. Thus, the exponential decline of prices is further bounded as follows: Rt ≤ (0.543)t−1 .


13

Proposition 6. The expected profit of Retailer 1 (resp., 2) is increasing in T . Further, it is bounded from below by

1 4

(resp., 0) if T = 1, and monotonically converges to about 0.2718 (resp.,

0.0803), as T → ∞. To measure the effect of the zigzag competition we compare the system’s profit under zigzag competition and under monopoly. The monopolist’s profit is increasing in T , is bounded from below by

1 4

when T = 1 and converges to

1 2

as T → ∞. Under zigzag competition, the system’s profit is

increasing in T , is bounded from below by

1 4

and converges to about 0.3522. Thus, the system’s 1 2

profit loss due to zigzag competition, at convergence, is

− 0.3522 ≈ 0.1477.

Proposition 7. The system’s profit loss due to the zigzag competition is bounded from below by zero (when T = 1), and it converges to about 0.1477 (when T → ∞), which represents a profit loss of about 29.6%. If we assume that in the zigzag competition model both retailers have equal probabilities to be visited by the consumer at the first period then we have: Proposition 8. The profit loss of a monopolist retailer due to the introduction of zigzag competition with a second retailer is bounded from below by

1 8

when T = 1 and converges to about 0.3239,

which represents a loss of about 64.78% of the profit. 2.4.

N Consumers

In this subsection we extend the model to N similar consumers, whose valuations are drawn independently from a uniform distribution on [0,1], and each of the retailers can satisfy the entire demand in the market, i.e., each has in stock N units of the good. The consumers do not all necessarily zigzag as a single group. That is, in the first period N 1 (resp., N 2 ) consumers visit Retailer 1 (resp., 2), with N 1 + N 2 = N , and, as before, the consumers maintain the zigzagging search pattern. Let us denote by Nti , i = 1, 2, the number of consumers who visit Retailer i in period j i , i 6= j = 1, 2, with N1i = N i . Let π(N t. Thus, Nti ≤ Nt−1 i ,N j ) denote the profit-to-go of Retailer i, t

t

i = 1, 2, in period t when there are Nti + Ntj consumers in the system in period t, and let Rti denote the respective price in that period. Since the consumers’ valuations may differ, it is possible that some consumers will make a Ni

purchase in a certain period while the others will continue to zigzag. Let Pn t , n = 0, ..., Nti , i = 1, 2, denote the probability that Retailer i, who is visited by Nti consumers, sells n units in that period. Ni

Thus, in period t, Pn t =

i Nti ! [Ft (Rti )]Nt −n [1 − Ft (Rti )]n . n!(Nti −n)!


14

Proposition 9. The profit-to-go of Retailer i, i = 1, 2, in period t when there are Nti + Ntj similar consumers in the system, such that Nti (resp., Ntj ) of them visit Retailer i (resp., j) in j 2 i i 1 i that period, can be expressed as π(N i ,N j ) = Nt πt + Nt πt , i 6= j = 1, 2, if t is odd, and π(N j ,N i ) = t

t

t

t

j 1 i i 2 1 2 Ntj πt2 + Nti πt1 , π(N i ,N j ) = Nt πt + Nt πt , i 6= j = 1, 2, if t is even, where πt and πt are the profit-to-go t

t

functions of Retailers 1 and 2, respectively, in the model with a single consumer. Thus, the resulting pricing policy is the same as in the single consumer case. In other words, the pricing policy is independent of the number of consumers in the system, as long as consumers are similar and each of the retailers can satisfy the entire market demand.

3.

Extensions to the Model

So far we have studied the model under the assumptions that consumers’ valuations are uniformly distributed and that the two competing retailers can satisfy the entire demand in the market. Hereby, we relax these two assumptions. In Section 3.1 we study the basic setting while assuming that the single consumer’s valuation follows a more general distribution, and in Section 3.2 we investigate the capacitated case. Section 3.3 briefly discusses other visit patterns of consumers between the two competing stores. 3.1.

Different Distributions

In the basic model previously considered we extend the valuation of the single consumer from uniform to a power distribution. That is, V , follows a power distribution, with a p.d.f. of the form (q + 1)V q , q > −1, 0 ≤ V ≤ 1. Thus, F (0) = 0 and F (1) = 1. Note that when q = 0, the power distribution coincides with the uniform distribution. To evaluate the effect of competition, we first develop the expressions for the monopolistic case (Section 3.1.1), then we elaborate the duopolistic case (Section 3.1.2), and we conclude with a short discussion on this effect (Section 3.1.3). 3.1.1.

Monopoly under Power Distribution One can show that when the monopolist is

facing a single consumer, his profit-to-go in each period is an affine function of the price that is set in the previous period. That is, πt = St Rt−1 , where St is a scalar, which can be expressed recursively as St =

q+1

1/(q+1) . (q + 2) (1 − St+1 )(q + 2)

(9)

This result is proved by induction, as in period t the profit-to-go is given by πt = Rt (1 − t t Ft (Rt )) + Ft (Rt )St+1 Rt = Rt (1 − ( RRt−1 )q+1 ) + ( RRt−1 )q+1 St+1 Rt . Taking the derivative w.r.t.

Rt to zero, we obtain Rt = q+1 R . (q+2)((1−St+1 )(q+2))1/(q+1) t−1

1 R . ((1−St+1 )(q+2))1/(q+1) t−1

Rewriting the profit results with πt =


15

By induction it can be shown that the sequence of {St } is backwards increasing in t and is bounded from above by S ∗ , which, using (9), is the value of S that solves S((1 − S)(q + 2))1/(q+1) = q+1 . q+2

Therefore, the sequence of {St } is a convergent sequence which converges to S ∗ . At convergence,

the value of the normalized sales-to-go coincides with the expected profit of the monopolist. For example, if q = 2 (resp., − 21 and 6), then the monopolist’s expected profit converges to 0.75 (resp., 1 3

and 0.875). Let Bt ≡

Rt Rt−1

=

1 . ((1−St+1 )(q+2))1/(q+1)

Noting that St =

q+1 B, q+2 t

one can express Bt in a

recursive way, Bt =

1 ((1 −

q+1 B )(q q+2 t+1

+ 2))1/(q+1)

.

The sequence of {Bt } is a convergent sequence, which converges, not surprisingly, to 1. It may be worth noting that for q = 0 the price decline is linear (i.e., it coincides with the case studied by Lazear) while for q < 0 (resp., q > 0) the decline is faster (resp., slower) than linear. 3.1.2.

Duopoly under Power Distribution In a duopoly, it can be shown, as before,

that the profit-to-go expressions in each period are affine functions of the price that is set in the previous period. For Retailer i (resp., j) who encounters (resp., does not encounter) the 1/(q+1) (q+2)/(q+1) j 1 1 consumer in period t, Sti = q+1 and Stj = St+1 . Let us q+2 (1−S i )(q+2) (1−S i )(q+2) t+1

t+1

denote by {sit } (resp., {sjt }) the sequence of normalized sales-to-go of the retailer who encounters 1/(q+1) 1 and sjt = (resp., does not encounter) the consumer in period t. Thus, sit = q+1 q+2 (1−sj )(q+2) t+1 (q+2)/(q+1) 1/(q+1) t sit+1 (1−sj 1 )(q+2) . Let Bt ≡ RRt−1 , and note that sjt = ( q+2 = (1−sj 1 )(q+2) si )q+2 sit+1 q+1 t t+1

and sit =

q+1 B. q+2 t

t+1

Thus,

1 Bt = q+1 (1 − q+2 (Bt+1 )q+2 Bt+2 )(q + 2)

1/(q+1)

.

By induction one can show that {Bt } is backwards increasing in t and that it is bounded from 1 1 below by ( q+2 )1/(q+1) and from above by B ∗ , which is real, satisfies ( q+2 )1/(q+1) ≤ B < 1, and solves 1/(q+1) B q + 2 − (q + 1)(B)q+3 = 1. Thus, based on the Monotone Convergence Theorem, {Bt } is

a convergent sequence, which converges to B ∗ . 3.1.3.

Effect of Competition under Power Distribution As q increases the center of the

valuation distribution shifts to the right. For values of q very close to −1 it is very likely that the consumer has a very low valuation for the product, and therefore the monopolist drops prices fairly quickly to increase the number of search points with low prices. Yet, the monopolist does not give up the opportunity of searching for the chance the consumer has a high valuation. As a consequence, we can see that, for example, when q = −0.5, even though there is 50% chance that the consumer’s valuation is less than 0.25, the first seven prices the monopolist sets in a 10-period


16

selling horizon are higher than 0.25 (Figure 4(a)). So the monopolist is trading the opportunity of a high gain with low probability and a low gain associated with high probability. On the other hand the competing retailers do not have the luxury of spending search points on high prices and due to competition they mark down prices fairly aggressively. For the same example, when q = −0.5, only the first two prices posted by the retailers exceed 0.25 (Figure 4(a)). As q increases and the center of valuation distribution shifts to the right, the consumer is more likely to have a high valuation for the product and, therefore, both the monopolist and the duopoly’s retailers set higher prices in each period. In a sense, both systems slow down the price search, but since the monopolist is not subject to competition, he can focus the search only on the “promising” area and, thus, be more successful. Figure 4 demonstrates the shifting of the price decline over a 10-period selling horizon for different q values and Figure 5 exhibits the expected system profit (both for a monopoly and a duopoly) as a function of the selling horizon for different values of q. In Figure 5 the difference between the plots is the system profit loss due to competition, which is increasing in T , the length of the selling horizon, but can be observed to decline percentage-wise as q increases. In other words, asymptotically, as q → ∞, the percent loss due to competition goes to zero. 1

1

1

1

0.8

0.8

0.8

0.8

Monopoly Monopoly 0.6

0.6

0.6

Monopoly

Duopoly

0.4

0.4

Duopoly 0.2

0.2

0

0

0

2

3

0.4 Duopoly

Duopoly

0.2

1

0.6

Monopoly

0.4

4

5

6

7

8

9

10

1

2

3

t, Period number

(a)q = − 12

4

5

6

7

8

9

10

0.2

0 1

2

3

4

5

6

7

t, Period number

t, Period number

(b)q = 0

(c)q = 5

8

9

10

1

2

3

4

5

6

7

8

9

10

t, Period number

(d)q = 50

Prices in a 10-period selling horizon when valuation follows a power distribution

Figure 4

Monopoly 1

1

1

1 Monopoly

0.8

0.8

0.8

0.8

0.6

0.6

0.6

0.6

Duopoly

Monopoly 0.4

0.4

Monopoly

0.4

0.4

0.2

0.2

0.2

0

0

Duopoly

Duopoly 0.2

Duopoly

0 1

2

3

4

5

6

7

T, Selling Horizon

(a)q = − 12 Figure 5

8

9

10

1

2

3

4

5

6

7

8

9

10

0 1

2

3

4

5

6

7

T, Selling Horizon

T, Selling Horizon

(b)q = 0

(c)q = 5

8

9

10

1

2

3

4

5

6

7

8

9

T, Selling Horizon

(d)q = 50

System profit for different selling horizons when valuation follows a power distribution

10


3.2.

17

Limited Capacity

So far we assumed that both retailers can satisfy the market. In this section we extend the model by analyzing the capacitated case. That is, we analyze the case where each of the retailers cannot satisfy the entire market demand, the consumers are similar and their valuations are uniformly distributed. First, in Section 3.2.1 we develop the monopolistic setting, as to have a base case for comparison. Then, in Section 3.2.2, we study the duopolistic competition. We also briefly investigate the role of information in the capacitated setting. 3.2.1.

Monopoly under Limited Capacity

The single unit case When a monopolist who is facing N similar consumers has only one unit of good in stock to sell over a selling horizon of T periods, his profit-to-go in each period can be expressed as an affine function of the preceding price. That is, it can be proved by induction that πt = St Rt−1 , where St is a scalar, which can be expressed recursively as St =

N (1−St+1 )1/N (N +1)(N +1)/N

. It can be further

proved: Proposition 10. The sequence {St } is a monotone convergent sequence as the number of periods, T , goes to infinity, while t goes to 1. Omitting subscripts we solve S = N (1 − S)−1/N (N + 1)−(N +1)/N , for which the solution is S ∗ = N , N +1

which represents the monopolist’s expected profit at convergence. This result is not surprising,

as at convergence the monopolist should obtain a profit which equals the expected value of the highest valuation all consumers. amongst N N N1 t t + RRt−1 St+1 Rt , it follows that Rt = (1−St+11)(N +1) Since πt = Rt 1 − RRt−1 . Figure 6 illustrates the prices set by a monopolist having a single unit in stock over a selling horizon of 20 periods for a varying values of N . K units in stock Now assume that the monopolist who is facing N similar consumers has K units of the good in stock. If in period t − 1 a price Rt−1 is set and not all goods are purchased, then all consumers who do not purchase the good have valuations below Rt−1 . Let PnNtt denote the probability that nt of the Nt consumers who visit the monopolist in period t have valuations which exceed the posted price Rt , i.e., PnNtt ≡

Nt ! t ( Rt )Nt −nt (1 − RRt−1 )nt . nt !(Nt −nt )! Rt−1

Note, that (i) if nt < Nt , then Nt+1 = Nt − nt , and

if nt ≥ Nt , then Nt+1 = 0, (ii) N1 = N , and (iii) R0 = 1. The monopolist profit in a two-period selling horizon is K−1

π=

X

n1 =0

K−n1 −1 X (PnN2−n1 n2 R2N −n1 ) + PnN1 n1 R1 + n2 =0

X

N −n1

n2 =K−n1

N X PnN1 KR1 , PnN2−n1 (K − n1 )R2N −n1 + n1 =K


18

1

N=50

Rt, Price

0.8 0.6

N=10 0.4

N=3

0.2

N=1

0 1

3

5

7

9

11

13

15

17

19

t, Period Number

Figure 6

Prices a monopolist sets in a 20-period model when K = 1 for varying values of N

where R2N −n1 is the price the monopolist sets in period 2 given that n1 consumers purchase the good in the first period. The optimal prices are found by solving this equation backwards, first for all second period prices, R2N −n1 , n1 = 0 . . . K − 1, and then for the first period price, R1 .

0.9

R1

0.8

0.8 0.7

0.7 R2

0.6

0.6 0.5

0.5

0.4

0.4

0.3

0.3 1

2

3

4

(a)Prices Figure 7

5

6

7

1

2

3

4

5

6

7

(b)Expected Profit

Prices a monopolist sets in a 2-period model and corresponding profit when K = 1 for a varying values of N

Due to the complexity of the expressions for the optimal prices, we do not provide a general closed-form solutions for R1 and R2 for any N and K. However, we can demonstrate their main characteristics. As Figure 7 illustrates that as N increases, both R1 , R2 , and the corresponding profit increase as well. Similarly, Figure 8(a) illustrates, as K increases, the optimal prices when N = 10, wherein the solid line represents R1 and the triangles below represent the second period prices in a deceasing order of leftover inventory. For example, when K = 5, the uppermost triangle


19

is the second period price when the monopolist has one unit remaining in stock (i.e., he sold 4 units in the first period), and the lowermost triangle corresponds to the case of 5 units in stock (in other words, nothing was sold in the first period). Observe that when K = N = 10, for any positive leftover inventory, the monopolist sets the same price. Note that as Figure 8(b) demonstrates, as K increases the expected profit increases as well, yet, the profit per unit stocked is decreasing. This reflects a decreasing incentive to stock inventory when there is a positive holding cost. Indeed, so far we have assumed zero wholesale price (or production cost) and zero holding cost. However, for wholesale prices larger than zero, the corresponding optimal stocking level may be less than N . For example, with N = 10 if the wholesale price is 0.2 (resp., 0.5), the optimal stocking level is K = 6 (resp., 3) with a corresponding profit of 0.31 (resp., 0.18) per unit.

0.9

3.5

0.8

3

Profit

2.5

0.7

R1

0.6

2 1.5

0.5

1

0.4

R2

0.5

Profit K

0

0.3 1

2

3

4

5

6

7

8

9

10

1

2

(a)Prices Figure 8

3

4

5

6

7

8

9

10

K, Units in Stock

K, Units in Stock

(b)Expected Profit

Prices a monopolist sets in a 2-period model and the corresponding profit when N = 10 for varying values of K

3.2.2.

Duopoly under Limited Capacity The capacitated duopoly case is fairly easy to

formulate but solutions are not simple to obtain as expressions involve polynomial of high degrees. Thus, let us first consider the simple two-period case, wherein each retailer has one unit of good in stock and is visited by one consumer in the first period. In other words, each of the retailers can satisfy half the market (and together they can satisfy the entire market). The expression of the expected profit of Retailer i is given by π i = (1 − R1i )R1i + R1i R1j (R2i (1 −

R2i )), R1j

i 6= j = 1, 2,

where the first term, (1 − R1i )R1i , is the expected profit in the first period when Retailer i sets a price R1i , and the second term reflects the conditional expected profit in the second period. Specifically,


20

R1i R1j is the probability that Retailer i is not out of stock and that the consumer who has visited Retailer j in the first period has not purchased the good, and R2i (1 −

i R2 j

R1

) is the corresponding

expected profit given that event.

√ √ Solving backwards, we find that R1i = R1j = 4 − 2 3 ≈ 0.535 and R2i = R2j = 21 R1i = 2 − 3, √ with a corresponding expected profit of 28 − 16 3 ≈ 0.287 for each of them. By comparison, the

corresponding prices set by a monopolist who is facing two consumers and is holding one (or two)15 unit(s) of good in stock are 0.736 and 0.425 (resp., 0.666 and 0.333) in the first and second period respectively, and the associated profit is 0.49 (resp., 0.666). Thus, duopoly competition results with lower prices than the monopoly, as expected, and a loss of 41% (resp., 13.8%) of the profit from a retailer’s (resp., system’s) perspective. Extending this setting to a three-period selling horizon, Retailer i’s profit is given by i

π =

(1 − R1i )R1i

+ R1i R1j R2i (1 −

R2i R3i R3i,m j i,m i j i i ) + R2 R2 R3 (1 − j ) + R1 (1 − R1 )R3 (1 − i ), R1 R1j R2

i 6= j = 1, 2,

Ri

where the first two terms are as before. The third term, R2i R2j R3i (1 − R3j ), is the conditional expected 2

profit in the third period if both retailers have not sold the good yet, and the last term, R1i (1 − R1j )R3i,m (1 −

i,m

R3

i R1

), reflects the expected profit in the third period when Retailer j sells the good in

the first period while Retailer i does not. In that case, Retailer i is not visited at all in the second period and may realize that he has become a monopoly, and the price that he sets in that case is denoted by R3i,m . To find the optimal solution for this setting, we solve the model backwards. R1i is found numerically (by using the best reaction functions of the retailers) to be about 0.6207. Correspondingly, R2i = 0.332, R3i = 0.166, R3i,m = 0.31, with an expected profit of 0.34 for each retailer. This is compared with prices of 0.809, 0.59, and 0.34 (resp., 0.75, 0.5, and 0.25) a monopolist sets over the three periods when he faces two consumers and has one (resp., two) unit(s) of good in stock, and an expected profit of 0.539 (resp., 0.75). Thus, when the horizon extends to three periods, duopoly competition keeps exerting pressure on prices and profits. Yet, this pressure is decreasing, and the loss incurred to the profit from a retailer’s (resp., system’s) perspective is 36.9% (resp., 9.3%), which is slightly lower than in the two-period horizon. Next, we consider the case wherein each retailer has one unit of good in stock and is visited initially by two consumers. That is, each retailer can satisfy a quarter of the entire demand and together they can satisfy half the market. In that case informational issues arise. 15

We provide these two comparisons as to provide retailer’s perspective (who can satisfy half the market) and a system’s perspective (which satisfies the entire demand). Retailer’s perspective reflects the case where an identical retailer is introduced into the monopolistic market and consumers are equally distributed between the two competing retailers in the first period. The system’s perspective represents a situation where both retailers could satisfy the entire demand together, so in a sense, the only “inefficiency” is due to competition.


21

No Sales Information When Retailer i cannot observe the sales at other location, he sets a price R1i in the first period which leads to an expected profit of R1i (1 − (R1i )2 ) in the first period. With probability (w.p.) (R1i )2 , Retailer i doesn’t sell at all in the first period and sets a price R2i in the second period. W.p. (R1j )2 Ri

two consumers arrive and w.p. 1 − ( R2j )2 at least one of them buys the good. If only one consumer 1

arrives to Retailer i, then w.p. 2R1j (1 − R1j ) his valuation is below R1j and he buys the good w.p. 1−

i R2 j

R1

, and w.p. (1 − R1j )2 his valuation exceeds R1j and he buys the good w.p. 1 (if R2i ≤ R1j ).

Assuming R2i ≤ R1j (and this can be verified to hold), the profit of Retailer i (after simplifying) is given by: π i = R1i (1 − (R1i )2 ) + (R1i )2 R2i

(R1j )2 − (R2i )2 + 2(1 − R1j )(R1j − R2i ) + (1 − R1j )2 . q

Solving backwards, we find that R2i = 13 (2R1j − 2 +

4(R1j )2 − 8R1j + 7) and we find numerically

that R11 = R12 ≈ 0.657. Thus, R21 = R22 ≈ 0.393 and each retailer’s expected profit is about 0.471. These prices are lower than the prices set by a monopolist holding one (resp., two) unit(s) of good in inventory and facing four consumers, in which case the prices the monopolist sets are 0.809 in the first period and 0.54 in the second period (resp., 0.74 in the first period, 0.43 and 0.466 in the second period if he sells nothing or one unit in the first period, respectively). Yet, these prices are higher than in the previous case wherein each retailer was visited by a single consumer. Consequently, the loss incurred to the profit from a retailer’s (resp., system’s) perspective is 27.2% (resp., 11.1%), which is dramatically lower than the profit loss when each retailer was visited by a single consumer, which was 41% (resp., 13.8%). With Sales Information We further assume that each retailer may observe the sales level at the other retailer. As each retailer has one unit in stock it means that they simply need to observe whether the other retailer is out of stock or not. Now, a retailer may set different prices in the second period, based on the i i observation made at the end of the first period. Let R2(1) (resp., R2(2) ) denote the price Retailer i

sets in the second period if only one consumer (resp., two consumers) visit(s) him in the second period. Assuming R2i ≤ R1j (and this can be verified to hold), the profit of Retailer i is given by: i i π i = R1i (1 − (R1i )2 ) + (R1i )2 R2(2) (R1j )2 − (R2i )2 + R2(1) 2(1 − R1j )(R1j − R2i ) + (1 − R1j )2 . i Solving backwards, we find that the second period prices are R2(1) =

first period price,

R1i ,

j

R1 +1 4

i and R2(2) =

√ j 3R1 . 3

The

is approximately 0.658. This price is only marginally higher than the price

set by the retailers in the case of no sales information. The corresponding second period prices are


22

0.414 if one consumer arrives and 0.3799 if two consumers arrive, and the corresponding profit is 0.4714. The value of information in that case increases the expected profit only by about 0.04%. Additional observations are presented in Table 2 below. 3.2.3.

Effect of Competition under Limited Capacity Let us briefly make several qual-

itative observations regarding the effect of competition in the most elementary capacitated case wherein each retailer has only one unit of good in stock. We have observed that, similar to the uncapacitated case, competition in the capacitated case dramatically suppresses prices, which results with a substantial decline in profit. As the number of consumers in the system increases, while inventory levels are fixed, the competitive pressure diminishes, and as a result prices are set closer to the monopolistic case and correspondingly, the profit loss is lower. Also, as the selling horizon extends, the competitive pressure declines. Table 2 below summarizes prices that are set and profit per consumer achieved in a two-period selling horizon. The first (resp., second) monopolistic observation in each case is to provide the retailer’s (resp., system’s) perspective, as was discussed earlier. Indeed, here we have used an elementary model, wherein each retailer has only one unit of good in stock, but it can be extended to other situations where retailers have more than one unit in stock.

a

Setting

R1i

R2i

Profit per unit

Two Duopolya Consumers Monopoly: 1 unit in stock in the system Monopoly: 2 units in stock

0.535 0.736 0.66

0.268 0.425 0.33

0.287 0.49 0.33

Four Duopoly: no sales info. Consumers Duopoly: with sales info. in the system Monopoly: 1 unit in stock Monopoly: 2 units in stock

0.657 0.658 0.809 0.74

0.392 0.414, 0.38 0.54 0.466, 0.43

0.4712 0.4714 0.647 0.53

Six Duopoly: no sales info. 0.726 0.478 Consumers Duopoly: with sales info. 0.726 0.5, 0.457 in the system Monopoly: 1 unit in stock 0.849 0.614 Monopoly: 2 units in stock 0.7911 0.698, 0.644

0.576 0.577 0.728 0.634

All duopoly cases refer to the single unit in stock at each retailer. Table 2 Summary of the capacitated case in a two-period selling horizon


3.3.

23

Other Visit Patterns

Our zigzag competition model assumes that if a consumer visits one of the retailers and observes a price that exceeds his valuation, then in the ensuing period he visits the competing retailer. In Granot, Granot and Mantin (GGM) (2006) we generalize this visit pattern and acknowledge that a consumer may return to the same retailer in the ensuing period even if the current price he observes exceeds his valuation. Explicitly, we assume therein that the consumer returns to the same retailer in the ensuing period with probability P or switches to the competing retailer with the complementary probability 1 − P , where P reflects market conditions or consumer’s experience at the store. In GGM (2006) it is found, among other results, that the exponential decline of prices is quite robust as it holds for fairly general visiting patterns by consumers and when the retailers are far from being symmetric.

4.

Identical Consumers

In this section we study pricing schemes when all consumers are identical, with a common valuation uniformly distributed on [0, 1] and the retailers cannot necessarily satisfy the entire demand. Let us consider only the symmetric case. That is, we assume that each retailer is visited by N consumers in the first period and has K units of goods in stock. We further assume retailers have sales information, i.e., they can observe whether consumers have purchased the good or not at the competing retailer. We find that in the identical consumers case there may exist only non-symmetric pure strategy SPNE as well as a continuum of equilibria. 4.1.

Uncapacitated Duopoly Two-Period Selling Horizon Assume that R1i ≤ R1j , and 2N ≤ K. Then, the profit

4.1.1.

functions for Retailers i and j can be written as π i = N R1i (1 − R1i ) + (R1j − R1i )R1i N + R1i N R2i (1 − i R2 i ), R1

and π j = N R1j (1 − R1j ) + R1i N R2j (1 −

j

R2 i R1

). Solving backwards, we find that R2i = R2j = 12 R1i .

Solving for the first period, we have R1i = min(R1j ,

j

2(1+R1 ) ) 7

and R1j = max( 12 , R1i ). Thus, we have

that the pure-strategy SPNE profile is R1i = 73 , R1j = 12 , and R2i = R2j =

3 . 14

Correspondingly, the

profits of the retailers are shown in Table 3. Obviously, the symmetric case in which R1i = 12 , R1j = 37 , R2i = R2j =

3 14

is also an SPNE. 1 2

Retailer j sets

Table 3

Retailer i sets

1 2

Retailer i sets

3 7

5 N 16 9 N 28

3 7

5 ≈ 0.312N , 16 N

≈ 0.32N ,

29 N 98

≈ 0.296N

Two-period expected profits for the two retailers

29 9 N , 28 N 98 57 N 196

57 ≈ 0.291N , 196 N


24

In view of Table 3 we can conclude: Corollary 2. In an uncapacitated two-period selling horizon with identical consumers, there exists precisely two pure-strategy SPNE both of which are non-symmetric. Recall that in the corresponding uncapacitated case when consumers are similar, the purestrategy SPNE is unique and symmetric, and is such that each retailer sets the price of the first period, and

1 4

1 2

in

in the second period. Thus, on average, a duopoly system facing identical

consumers reduces the first period price by 7%. Correspondingly, the system’s profit drops from 2 N 3

(since it faces 2N consumers) to

121 N, 196

which represents a decline of almost 7.4%. Thus, in

the uncapacitated case, identical consumers increase the competitive pressure on the two retailers. Thus, we have: Corollary 3. In an uncapacitated two-period selling horizon equilibrium prices and corresponding profits are lower when consumers are identical 4.1.2.

Longer Selling Horizons We can extend the above analysis to longer selling horizons.

However, with longer selling horizons the number of possible SPNE grows quickly. Let us consider the two possible such SPNE wherein (i) one of the retailers always sets the lower price while the other sets the higher price and (ii) the retailers alternate in these roles. i Retailer i sets the lower price in all periods The profit functions in period t are πti = N Rti (Rt−1 − j i i Rti ) + (Rtj − Rti )Rti N + St+1 N (Rti )2 and πtj = N Rtj (Rt−1 − Rtj ) + St+1 N (Rti )2 , where Sti (resp., Stj )

is the normalized profit-to-go of Retailer i (resp., j) in period t. We can show by induction that Sti = and

j

i i 16−16St+1 +4(St+1 )2 +9St+1 j 9 i and S = . The corresponding prices are Rti = 4(2−S3 i ) Rt−1 i i t 16(2−St+1 ) 16(2−St+1 )2 t+1 3 i Rtj = 21 Rt−1 , and the price ratios are Bti = 4(2−S3 i ) = 8−3B and Btj = 21 . It can be shown i t+1 t+1

that {Bti } is a convergent sequence for t < T − 1, which converges to 0.451. Therefore, prices decline exponentially at convergence (that is, T → ∞), R1i = 0.451, R1j = 0.5, π i = 0.338N and π j = 0.3139N . This represents a system profit loss of 34.8% due to competition, when compared with the monopolist’s profit, which converges to N if he holds 2N units in stock. Recall that the price ratio sequence in the corresponding uncapacitated case when consumers are similar is also backwards increasing and bounded from below by 12 , which is greater than or equal to Btj (which in turn is greater or equal to Bti ), for any t, and strictly greater for t < T − 1. This leads us to the following conclusion: Corollary 4. In the uncapacitated case, if consumers are identical and the same retailer constantly sets the lower price at each period, the resulting prices at each period are lower or equal to the prices set when consumers are similar.


25

Alternating Lower Price The analysis for that case is similar to the preceding case. Indeed, it can be shown that, at convergence, the lower (resp., higher) price in the first period is 0.445 (resp., 0.5), Corollary 4 holds in this case as well, and prices decline exponentially. 4.2.

Capacitated Duopoly

Let α ≡

K . N

Theorem 2. In a capacitated duopoly with two-period selling horizon and identical consumers, when α >

3 2

2 3

there is a continuum of symmetric pure-strategy SPNE; when α =

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